Tempi
Tempi (singular: tempo) play an important role in many abstract strategy games including Chess, Checkers, and Mancala. Tempo refers to a "turn" or single move. When a player achieves a desired result in one fewer move, one "gains a tempo" and conversely when one takes one more move than necessary one "loses a tempo". Similarly, when one forces one's opponent to make moves not according to the initial plan, one "gains tempo" because the opponent wastes moves. A move that gains a tempo is often called a move "with tempo". The object of games with an absolute goal is to achieve this goal faster than the opponent, e.g. to mate your opponent's king before he is able to mate your own one (in Chess), to achieve a particular endgame position faster than your opponent (in Salta), to be the first to connect the opposite edges of the board by a chain of counters (in Hex). Even in games which are decided by the number of points a player conquers, tempi are often very important. In the East Asian game of Go a semeai is a capturing race, in which groups of stones strive to capture each other. Mancala games are characterized by a large variety of goals. Some have absolute goals such as moving last (e.g. Imbelece, Hawalis) or capturing all the opponent's seeds of his front row (e.g. Bao la Kiswahili, Kisolo). Others have a relative goal, that is, to be the player who captures more seeds (e.g. Congkak, Kalah, Oware). The concept of tempi is known to native players as well. It is called "bidoua" in Songo and has been discussed in Kazakh books about Toguz Kumalak. Games With Scoring System In many mancala games, which have a scoring system, the player who could move last wins the seeds that remained in the opponent's holes. It is therefore very important to not be forced to "feed" the opponent, that is to move seeds to his row. Example 1: Oware South to move! Solution 3!/3/2/4/4!/5/3!/6/1! - South wins 26:22. Wrong: 2? (loses a tempo)/3/3/4/5/5/4/6/6/ - Endless Cycle! The game is a draw (24:24). Example 2: Toguz Kumalak A) Case 1: Basic Position Let's assume that just one hole is occupied (condition 1 - but see note below) and that its contents does not reach to the opponent's row (condition 2). Furthermore, the opponent's hole 1 may not be a tuzdik (condition 3 - otherwise see B) below)). Then the number of tempi can be calculated as follows: 1) If s'' < 3: T = s x d 2) If ''s ≥ 3: T = ½ s x (2d - s + 1) + 1 (T stands for the number of tempi; s for the number of seeds and d for the distance between the occupied hole and the opponent's row) Note: These formulas can be applied to any board position in which the holes can be emptied one after another without making the contents of any other hole > 2. B) Case 2: Opponent's 1 is a Tuzdik Let's assume that just one hole is occupied (condition 1 - but see note above) and that its contents does not reach farther than the opponent's tuzdik in hole 1 (condition 2). Then the number of tempi can be calculated as follows: 1) If s'' < 3: T = s x (d + 1) 2) If ''s ≥ 3: T = ½ s x (2d - s + 3) + 1 C) Case 3: Opponent's 2 is a Tuzdik Let's assume that just one hole is occupied (condition 1 - but see note above) and that its contents does not reach farther than the opponent's tuzdik in hole 2 (condition 2). Then the number of tempi can be calculated as follows: 1) If s'' < 3: T = s x d 2) If ''s ≥ 3: T = ½ s x (2d - s + 1) + 1 In other words, the formulas are exactly the same as in case 1. How many tempi has South? - No tuzdik. Solution: T = T(1) + T(2) + T(3) = : = x 3 x (2 x 2 - 3 +1) + 1 + x 5 + x 8 x (2 x 7 - 8 + 1) + 1 = : = x 2 + 1 + 10 + x 7 + 1 = : = 4 + 10 + 29 = : = 43 Games With Absolute Goals Example 1: Waurie Waurie is unusual in that the object of the game is to empty your own rows. Therefore the players try to vacate their holes as fast as possible and to "feed" their opponent. That is they try to reduce their own tempi while giving their opponents new tempi. South to move and win! Solution: 3! (reduces one tempi)/3/4/4/5!/5/6!/1/2/2/3/3/4/4/5/6-5/6 Example 2: Hawalis The object of Hawalis is to be the last player able to move - in other words, to have tempi. North to move! Solution: E!/n/G/a/H/b/I/c/Jx3/e/F/f/G/g/H/i/Kx1/h/Lx1 - South has no tempi. But not: Gx3/k/E/i/F/j/H/lx1/G/mx1 - North lost all tempi. External Links *Tempo in Chess (Wikipedia) *Tempo in Checkers (video) *Tempo in Go (Sensei's Library) References ;Meka Obam, J.-M.: Le Jeu du Songo: Reflet Du Social. Éditions L'Harmattan, Paris (France) 2008. ;Sharipov, C. & Shotayev, M.: Toguz Kumalak: Igra Intellektualov. Alma-Ata (Kazakhstan) 2006. Copyright © Ralf Gering Under the CC by-sa 2.5 license. Category: Mathematics